I remember being a kid and reading Harry Potter and started noticing that they had a character called "Mad-Eye" Moody and they would call people mad when they thought they were going crazy. This was new way to use this word for me and I always thought it was weird.
When (at least in the US) did mad become angry?
This is sense 6b in the Oxford English Dictionary under "mad" (adj.):
Angry, irate, cross. Also, in weakened sense: annoyed, exasperated (with †against, at, with, etc.). Now colloq. (chiefly N. Amer.) and Brit. regional.
The earliest citation for this sense is from 1400, and there are citations from texts written in each century thereafter. This one from William James is interesting:
W. James Varieties Relig. Experience xi. 264 "He can't ‘get mad’ at any of his alternatives; and the career of a man beset by such an all-round amiability is hopeless."
The quotation marks with "get mad" suggest that James perceived the phrase as colloquial (though not necessarily the adjective by itself in this sense).
In sum, the adjective "mad" has always had "angry" as a possible sense. The OED citations for this sense do seem to shift away from British writings toward American in the 20th century, but that isn't evidence that this sense is no longer current in British English.